A Primer for Preproblem Ponderings: Anticipating the Answer

A Primer for Preproblem Ponderings: Anticipating the AnswerAuthor(s): Karen Singer Cohen and Thomasenia Lott AdamsSource: The Mathematics Teacher, Vol. 97, No. 2 (FEBRUARY 2004), pp. 110-115Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20871524 .

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Karen Singer Cohen and Thomasenia Lott Adams

A Primer for Preproblem Ponderings: Anticipating the Answer

w

If you hand an answer

to someone, what type of objects will

you be

handing over?

hat is the initial work of the problem solver? It is to comprehend the important components of the prob lem: the unknown, the data, and the conditions

describing the unknown (P?lya 1957). If we assume

that a problem is a task for which the solution method is not known in advance (NCTM 2000, p. 52), the problem solver is then challenged to

navigate the gap between the given data and the un

known. Many students falter early in the problem solving adventure and are frustrated by not know

ing immediately what steps to take. Others perform calculations without taking the time to consider what the problem demands and are thus unable to determine a wise solution path. "Preproblem pon

derings" can build a foundation for the problem solving process. We present a strategy for prompting the pre

problem ponderings of students: We call it "antici

pate the answer." We encourage teachers to intro duce this strategy, or their own version of it, as a

segment of the "broad repertoire of problem-solving (or heuristic) strategies" that the National Council of Teacher of Mathematics (2000, p. 335) suggests equips students for problem solving.

ANTICIPATE THE ANSWER What is "anticipate the answer"? The goal of this

preproblem pondering strategy is simple: trying to

anticipate what the final answer will look like. It includes both the form of the answer and the an

swer's relationship to the conditions of the problem. Specifically, we expect students employing this heuristic strategy to identify the facts of the problem, identify the question or questions to be answered, and then respond to the following questions:

If you hand an answer to someone, what type of

objects will you be handing over?

What will you guarantee about the answer?

The first question focuses attention on the type of objects to expect. Although the type of object is commonly a number, it can vary widely. We encour

age students to include units or assign a variable name where relevant. Since we assume that all

answers will have an accompanying explanation, we generally do not list that explanation as an

object type here. In addition to helping students

begin parsing and considering the problem, this

question frames mathematics as a world of objects. The second question, What will you guarantee

about the answer, demands thought about the condi tions that the answer will satisfy. A student might provide a condition that is equivalent to the initial

problem statement, thereby containing enough information to describe an answer completely; or he or she might simply list a characteristic or property that is assumed or known to be true about the answer. Although the condition is usually a mathe matical one, the guarantee could simply relate the answer to the problem's application context, such

as, the item is something that can be purchased at a sporting goods store. The guarantee statements are not required to correspond one-to-one to the

object-type statements, so students can exercise

flexibility in their thinking without the constraint of keeping the two parts of the process completely connected. Sample responses to both of the ques tions appear in table 1.

Although this exercise asks students to look past the solution process to the final answer, focusing on those aspects of the answer can aid a student in

making the transition into thinking about the

process for determining such an answer. We

encourage students to supply as many details as

possible in their responses, even stating the same

guarantees different ways, since doing so may

Karen Cohen, [email protected], teaches mathematics

education at the University of Florida, Gainesville, FL 32611-7048. Her publications address mathematics prob lem solving, random graph theory, and designing mathe

matics enrichment programs. Thomasenia Adams,

[email protected], teaches undergraduate and graduate mathematics methods courses in an integrated mathemat

ics, science, and technology teacher preparation program at the University of Florida, Gainesville, FL 32611-7048. She is interested in algebra, geometry, measurement, multi cultural mathematics, and curriculum integration.

110 MATHEMATICS TEACHER



illuminate additional important relationships. Although a primary goal of the method is to help students narrow the focus of their problem solving toward obtaining a specific product, the experience of restating conditions several ways can help them avoid limiting too much how they consider the

problem. Once launched into their problem-solving process, we expect and encourage students to try a

variety of avenues.

To illustrate the application of "anticipating the

answer," we first consider two problems chosen from recent NCTM publications. For each, we offer

sample responses that are appropriate and we describe how they might lead to fruitful results. We then share two classroom examples of problems and the resulting anticipate-the-answer responses constructed by students who were learning the

technique.

PROBLEM 1 I have a faithful dog and a yard shaped like a right triangle. When I go away for short periods of time, I want Fido to guard the yard. Because I don't want him to get loose, I want to put him on a leash and secure the leash somewhere on the lot. I want to use the shortest leash possible, but wherever I secure the leash, I need to make sure the dog can reach every corner of the lot. Where should I secure the leash? (NCTM 2000, p. 354)

Possible responses that anticipate the answer

1. If you hand an answer to someone, what type of


One point A description of how to find a point A diagram A demonstration with a triangle and a piece of string Directions for locating a specific place in the

yard

2. What will you guarantee about the answer?

It has the least maximum distance to the ver tices of the triangle. It shows how to minimize the greatest dis tance that needs to be traveled to reach a corner.

It is not too far from any one corner.

It allows for the shortest leash possible that still reaches every corner of the lot.

It is equidistant from all vertices.

One of the main challenges of this problem is

understanding the role that the leash plays. In re

stating the problem, these sample responses show different choices of vocabulary and a choice between

_TABLE 1_ _ Anticipate the Answer__

KeyWords_Sample Responses _

Type of objects A number An angle measure in degrees Two distances in centimeters A price in dollars A rule A generalization A function A graph Three equations A diagram or drawing A triangle Four points Directions for a geometric construction A trigonometric expression A map An assignment of numbers to spaces A list of corresponding congruent parts A description A proof A recommendation

Guarantee When we add them together, we get 10. The lines are perpendicular. It is the shortest route that reaches all the stations. It is the y-intercept of the line y = 1.9x + 4.

Every class on the list is matched with a room. Itis the amount of money that Tai spends in a month. It is the minimum distance that the photographer should be from the model.

It proves that the function is continuous. It represents the future value of an investment.

continuing to use the context of the yard or abstract

ing to the right triangle. Thus, students give the

object type different names and express the guar antee in varying ways. A single person or several different people could have given these responses. The last response makes a leap to an equivalent problem that would probably only occur to students after they had played with the initial problem (for example, in an interactive geometry software envi ronment). We encourage students to add such alter native statements, as they recognize them, to their initial response throughout the problem-solving process.

PROBLEM 2 A subdivision is being placed on a piece of land 1000 m by 1500 m. Within that land, an access road of uniform width forms the border of the subdivision. The area of the inner rectangle of houses and parks is to be at least 1.35 million m2 to accommodate the planned homes and parks. What is the largest width that can be set aside for the access road?

(modified from NCTM 2001, p. 8)

What will you guarantee about the answer?

I

Vol. 97, No. 2 ? February 2004 111



Preproblem

analysis can be a

worthwhile and fruitful

approach

An evolving list of responses that anticipate the answer

1. If you hand an answer to someone, what type of


a width a; (in meters)

2. What will you guarantee about the answer?

It is the largest width possible for the border around the subdivision, using the facts that

the outer rectangle is 1000 m by 1500 m and that the inner rectangle must be at least 1.35

million m2.

It makes the inner rectangle exactly 1.35 mil

lion m2.

When we subtract 2w from each dimension of

the plot of land, we obtain an inner rectangle that is 1.35 million m2.

When we subtract 2w from each dimension of

the plot of land, we obtain an inner rectangle that is 1,350,000 m2.

(1000 - 2m;)(1500 - 2w) = 1,350,000.

Since the original problem asks, "What is the

largest width ... ?" the object type is a width (or

distance, number, and so on). This problem solver

also makes progress in attending to the problem by

adding a variable name (w) and specifying that the

final answer should be in meters. We imagine that one problem solver writes the ensuing list of guar antees, further specifying the guarantee ideas each

time, until he or she reaches an algebraic represen tation of the problem. The first guarantee states

the numerical part of the problem concisely, drop

ping some elements of the story (for example, hous es and parks). The second iteration notes that the

problem can be viewed as one about equality, rather

than maximization. In the third statement, the geo metric and computational relationships of w to the

other numbers are made precise. In the fourth

The decrease in value over time is called depreciation For example, the percent of

depreciation of the Oldsmobile 88 from 1986 to 1990 is 46 percent.

If a buyer ranked percent of depreciation as the most important factor in choosing a new

car, which of the four vehicles would you recommend? Explain your response.

Place Your Anticipate the Answer statement here, using the following format.

The answer will be [object types], such that [condition].

SUCK Vr<Or g?4* \V Vy^> *?nsfr \eO^V?t^?tWO^^n ?

Fig. 1

statement, the units of inner area are converted from millions of meters to meters. The fifth state ment is a completely algebraic representation of the condition that relates the unknown to the data.

CLASSROOM EXAMPLE 1 Four points?A, B, C, and D?occur in that

sequence along a line. Taking them in pairs, the

six distances between these points, listed in

order of size, are 16, 23, 37, 39, 60, and 76. If

AB > CD, find the length of CD. (Adapted from NCTM 1991)

Students' responses that anticipate the answer

Below are representative examples of preservice teachers' responses to preproblem pondering by an

ticipating the answer. They composed the responses after their initial introduction to the technique.

The answer will be?

the length between C and D such that this dis tance is shorter than the distance between A

and B.

a whole number such that it is 16, 23,37, 39, or

60; but it cannot be 76, because one of the other measurements must be larger than the answer.

The first of these responses was by far the most common type in our class. It represents only a

rephrasing of the most obvious subcondition in the

problem (AB > CD). However, writing it does remind the students that it must be satisfied. This

problem has multiple subtle conditions that

emerge during the interpretation process, and the

second response begins to access them. It applies the AB>CD condition specifically in the context of

the numbers given in the problem. The teacher can

make explicit for students the comparative values

of such response types.

CLASSROOM EXAMPLE 2 Use the given tables of car trade-in values (provid ed to the students) to answer the following ques tion: If a buyer ranked percent of depreciation as

the most important factor in choosing a new car, which of the four vehicles would you recommend?

(inspired by SIMMS 1996)

A student's response that anticipates the answer

The response in figure 1, written by a seventh

grade student, shows an important step made in

reading and understanding a complex problem.

Although the word most appears in the problem statement, it is indeed the least depreciation that

is sought. In anticipating the answer, the student

realized this fact at the start of the problem, before

she launched into calculations.




IMPLEMENTATION IN THE CLASSROOM

We suggest teacher-facilitated whole-class discus sion for introducing the anticipate-the-answer strategy, followed by individual practice. Students need the opportunity to try the technique on a vari

ety of problems, with ongoing feedback and discus sion after the initial introduction of the technique. The amount of support needed varies with the level of the students. The concrete concept of an object type for the answer seems to be easier for students to comprehend than the idea of a guarantee about the answer. A possible lesson plan for introducing the strategy follows.

1. The teacher provides students with the two

anticipate-the-answer questions and sample responses for each one (as in table 1), in a place where students can continue to refer to them. He or she encourages students to add to the list of

sample responses. For more practice with guar antees, the teacher can suggest a particular object type and ask students for a potential guar antee that makes sense for the object. For exam

ple, given that the answer will be a distance, one student wrote that he might guarantee that it "is

greater than 100 kilometers." 2. The teacher next illustrates to the class his or

her own application of the technique to a problem, sharing with the students his or her thinking process and working all the way through a solu tion. He or she refers back to the responses ex

plicitly. The teacher may add to them or update them while completing the solution and can check the final results against them.

3. When the students are comfortable with the

anticipate-the-answer questions, the teacher pre sents them with another sample problem and asks them to write responses. We often add oral details to help students connect with the ques tions, such as, "if your boss at work needs a solu tion to this problem, what type of objects will you be handing over to him or her?"

4. The teacher collects sample student responses for the class to analyze together to see whether

they are appropriate and whether they suggest any particular process for solving the problem. The teacher solicits further refinements or restatements of the guarantees. With the help of the class, the teacher completes a solution to the

problem. 5. The teacher repeats the previous two steps for

several problems, without necessarily continuing beyond the anticipation step. He or she may assign completing some of these problems as homework and can then discuss at the next class session whether the experience of anticipating the answer in class affected students' experience

in working on those problems. 6. Tb assess individual students' understanding of

the technique, the teacher can require students to anticipate the answer as an initial part of their solution on some problems that he or she will collect, lb help students assess any benefits for their own problem solving, the teacher can ask students to supply written details about their process throughout their work on the prob lem, including what happened between the time that they wrote the response and the time that

they moved further into a solution, as well as

indicating when (or if) they referred back to their response. It is an opportunity for students to practice metacognitive analysis. In reading their analyses, the teacher may be able to point students to additional connections with their ini tial response. However, some students may natu

rally find the anticipate-the-answer technique more useful than others do, and the teacher should not grade students on how much they used the technique beyond the required initial stage.

When practicing the technique with students, the teacher will find that some of their guarantees are ones that are required by the problem, whereas others are personal ones that they are making on their way to their own solution. Figure 2 shows the work done by an eighth-grade student in antici

pating the answer for the following problem:

Find three points that all lie on one line.

Student Shed 1 for Anticipating the Answer

Please read this problem, and then respond to the two questions that help you anticipate some details about the answer to the problem.

Find three points that all lie on one line.

Ifyouhandan answer for this problem to someone, what type of e?jectfs) will you be handing over? . .

What will you guarantee about an answer to this problem? (List as many guarantees as

you can.)

U wor<WrU>of ? A p?;*t will I* P"**r?*J h VU 6*>nl?*fe* "*<r fU/o fH coorcWrcb o\ *ft?

p?iAf vor fi

fr*?t>

Fig. 2

Vol. 97, No. 2 ? February 2004 113



The strategy aids students

who would otherwise not

know how to

begin

The student's statement that "the coordinates of one point will be proportional to the coordinates [of] the other two" is not uniformly true for all lines.

However, this problem requires her to choose a par ticular line, and her guarantee is valid for the line that she chooses. The difference between the two

types of guarantees should be explicitly communi

cated, perhaps by putting the personal ones in

parentheses or labeling them with the initials of the problem solver.

We do not recommend requiring that students use this technique for every problem. Rather, stu dents should have enough practice with it that they are comfortable using it when they choose to do so. The most important thing that we convey to stu dents is our attitude that preproblem analysis can be a worthwhile and fruitful approach, whether we use the anticipate-the-answer technique or a simi lar one. Just as looking back can reveal more about a problem and its answers, looking ahead can be

beneficial, as well.

THEORETICAL SUPPORT Schoenfeld highlights the critique that one of the difficulties in applying the types of heuristic strate gies that P?lya suggests has been that "the charac terizations of them were descriptive, rather than

prescriptive" (1992, p. 353). The anticipate-the answer technique helps systematize steps that can be taken to address P?lya's initial deep questions about the structure of the problem. It addresses issues that are relevant to any problem and is fairly simple to implement without extensive problem solving experience.

In "The Language of Quantification in Mathe matics Instruction,* in Developing Mathematical

Reasoning in Grades K-12, Susanna S, Epp laments that although "the abilities to rephrase statements in alternative, equivalent ways, to rec

ognize that other attractive-looking reformulations are not equivalent... are crucial mathematical

problem-solving tools,... numerous studies show

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that students do not acquire these abilities sponta neously" (1999, p. 190). Writing responses that

anticipate the answer provides students with a structured opportunity to apply precise use of mathematics vocabulary and logic, as well as an

opportunity to practice reformulating a problem. In

particular, the method encourages proper use of definitions. For example, if a problem asks for a

slope, then the object type of the answer could be a

slope, a number, or a ratio, but it could not be a

pair of coordinates. Our quest to encourage students to envision the

answer's form is related to the strategy of estimat

ing an answer for a numerical problem ahead of

time, an approach shown by De Corte to be an effective problem-solving strategy. He explains that the heuristic value of the estimating strategy is that it will "influence the proper solution process in the sense that the problem space is reduced. Such a reduction in the search space implies better insight into the nature of the problem" (1981, p. 8),

Writing down a name for the desired object type must immediately reduce the problem space for the

solver, as must identifying any conditions that the answer must satisfy. If students are prevented from

moving forward in analyzing a problem because

they are intimidated that the situation seems very "wide open," then this reduction should have bene fits for student attitude, in addition to any benefits for the student's mathematical vision, as applied to the problem.

By serving as an impetus for students to ponder their thoughts about a problem before?as well as

during?the problem-solving process, using the

anticipate-the-answer strategy also promotes metacognitions. Referring to and updating the lists of object types and guarantees while working on the problem allow a student to connect his or her work with the initial problem statement and help him or her assess progress. Metacognition is an

important component of intellectual growth, and research shows that students can develop their

metacognitive abilities through explicit instruction (Gollubetal. 2002).

CONCLUSION Anticipating the answer draws on skills of recogni tion, identification, and interpretation. It helps stu dents "enter" a problem, facilitating the transition from the initial phase of understanding a problem into the phase of working on a solution. It builds confidence by adding structure to a process that can be intimidating.

The strategy is intended to aid students who would otherwise not know how to begin. Yet, even

advanced problem solvers can find the approach useful on challenging problems, since it helps

] organize initial thoughts and starts a chain of




mental associations about the substance of the

problem. We observe that when students anticipate the

answer, they often remain more focused throughout the problem-solving process. This preproblem pon dering is a process that frames the probable answer in the problem statement in such a way that stu dents are empowered not to lose sight of the goal.

A great strength of this preproblem pondering strategy is its generality. It can be executed for any level of mathematics and for any mathematics con tent area. Once a student has had some guided practice in applying the technique, anticipating the answer can become a readily used tool in his or her

problem-solving tool box.

REFERENCES De Corte, Erik. "Estimating the Outcome of a Task as

a Heuristic Strategy in Arithmetic Problem Solving: A Teaching Experiment with Sixth-Graders." Paper presented at the annual meeting of the American Educational Research Association. Los Angeles: 1981.

Epp, Susanna S. "The Language of Quantification in Mathematics Instruction." In Developing Mathemati cal Reasoning in Grades K-12, 1999 Yearbook of the National Council of Teachers of Mathematics

(NCTM), edited by Lee V. Stiff and Frances R. Curcio, pp. 188-97. Reston, Va.: NCTM, 1999.

Gollub, Jerry P., Meryl W. Bertenthal, Jay B. Labov, and Philip C. Curtis, eds. Learning and Understand ing: Improving Advanced Study of Mathematics and Science in U.S. High Schools. Washington, D.C.: National Academy Press, 2002.

National Council of Teachers of Mathematics (NCTM). "February Calendar." Mathematics Teacher 84 (Feb ruary 1991): 114-17.

-. Principles and Standards for School Mathe matics. Reston, Va.: NCTM, 2000.

-. Navigating through Algebra in Grades 9-12. Reston, Va.: NCTM, 2001.

P?lya, George. How to Solve It. 2nd ed. Princeton, N.J.: Princeton University Press, 1957.

Schoenfeld, Alan H. "Learning to Think Mathemati

cally: Problem Solving, Metacognition, and Sense

Making in Mathematics." In Handbook of Research on Mathematics Teaching and Learning, edited by Douglas A. Grouws, pp. 334-70. Reston, Va.: NCTM, 1992.

SIMMS. Integrated Mathematics 1 (1). Needham Heights, Mass.: Pearson, 1996.

We wish to thank teacher Aharon Dagan and the administration at the Hawthorne Junior-Senior

High School, in Hawthorne, FL 32640, for allowing us to share this technique with their students.

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A Primer for Preproblem Ponderings: Anticipating the Answer